How to add movie to timeline and start in middle of movie
Below the Edit Strip tab, where you had set the Start Frame, there is the Strip Input tab.

If you want the strip to actually display a later frame on your frame 1, change the Start property below Trim Duration (hard) to your desired start frame of the clip.



Another more manual way of doing this.


Scrub forward to or pause on your desired start frame.
Press Shift + K to cut the clip into two.
Select the first (now redundant) clip and delete it by pressing X.
Select the clip which now starts on your desired frame and move it backwards to the start of the scene by pressing G and then dragging the mouse. Once in grab mode, you can also enter an offset (like "-60") with the keyboard.


Reference: Blender Manual Entry